A cocyclic construction of $S^1$-equivariant homology and application to string topology

Given a space with a circle action, we study certain cocyclic chain complexes and prove a theorem relating cyclic homology to $S^1$-equivariant homology, in the spirit of celebrated work of Jones. As an application, we describe a chain level refinement of the gravity algebra structure on the (negative) $S^1$-equivariant homology of the free loop space of a closed oriented smooth manifold, based on work of Irie on chain level string topology and work of Ward on an $S^1$-equivariant version of operadic Deligne's conjecture.


INTRODUCTION
Let M be a closed oriented smooth manifold and LM = C ∞ (S 1 , M ) be the smooth free loop space of M .In a seminal paper [3] (and a sequal [4]), Chas-Sullivan discovered rich algebraic structures on the ordinary homology and S 1 -equivariant homology of LM , initiating the study of string topology.In particular, there is a Batalin-Vilkovisky (BV) algebra structure on (shifted) H * (LM ) ([3, Theorem 5.4]), which naturally induces a gravity algebra structure on (shifted) H S 1 * (LM ) ([3, Section 6], [4, page 18]).The goal of this paper is to describe a chain level refinement of the string topology gravity algebra, and compare it with an algebraic counterpart related to the de Rham dg algebra Ω(M ).Along the way we also obtain results on the relation between cyclic homology and S 1 -equivariant homology, and an S 1 -equivariant version of Deligne's conjecture.
In spirit, this paper may be compared with work of Westerland [33].Westerland gave a homotopy theoretic generalization of the gravity operations on the (negative) S 1 -equivariant homology of LM , whereas we describe a chain level refinement.
Cyclic homology and S 1 -equivariant homology.The close connection between cyclic homology (algebra) and S 1 -equivariant homology (topology) was first systematically studied by Jones in [21].One of the main theorems in that paper ( [21,Theorem 3.3]) says that the singular chains {S k (X)} k≥0 of an S 1 -space X can be made into a cyclic module, such that there are natural isomorphisms between three versions of cyclic homology (positive, periodic, negative) of {S k (X)} k≥0 and three versions of S 1 -equivariant homology of X, in a way compatible with long exact sequences.
The first result in this paper is a theorem "cyclic dual" to Jones' theorem.As far as the author knows, such a result did not appear in the literature.
Theorem 1.1 (See Theorem 3.1).Let X be a topological space with an S 1 -action.Then {S * (X × ∆ k )} k≥0 can be made into a cocyclic chain complex, such that there are natural isomorphisms between three versions of cyclic homology of {S * (X × ∆ k )} k≥0 and three versions of S 1 -equivariant homology of X, in a way compatible with long exact sequences.
Jones dealt with the cyclic set {Map(∆ k , X)} k and the cyclic module {S k (X)} k , while we deal with the cocyclic space {X × ∆ k } k and the cocyclic complex {S * (X × ∆ k )} k .It is in this sense that these two theorems are "cyclic dual" to each other.In the special case that X is the free loop space of a topological space Y , Theorem 1.1 may also be viewed as "cyclic dual" to a result of Goodwillie ([18, Lemma V. 1.4]).As does Jones' theorem, Theorem 1.1 has the advantage that it works for all S 1 -spaces.
The cyclic structure on singular chains plays no role in Theorem 1.1; what matters is the cocyclic space.Indeed, the main motivation for the author to seek for a result like Theorem 1.1 is to study the S 1 -equivariant homology of LM , using a novel chain model of loop space homology defined via certain "de Rham chains", introduced by Irie [20].
Deligne's conjecture.What is called Deligne's conjecture asks whether there is an action of a certain chain model of the little disks operad on the Hochschild cochain complex of an associative algebra, inducing the Gerstenhaber algebra structure on Hochschild cohomology discovered by Gerstenhaber [12].This conjecture, as well as some variations and generalizations, has been answered affirmatively by many authors, to whom we are apologetic not to list here.What is of most interest and importance to us is work of Ward [31].
Ward ([31,Theorem C]) gave a general solution to the question when certain complex of cyclic (co)invariants admits an action of a chain model of the gravity operad, inducing the gravity algebra structure on cyclic cohomology.Recall that the gravity operad was introduced by Getzler [16] and is the S 1 -equivariant homology of the little disks operad.So Ward's result can be viewed as an S 1 -equivariant version of operadic Deligne's conjecture ( [31,Corollary 5.22]).
The second result in this paper is an extension, in a special case, of Ward's theorem.To state our result, let A be a dg algebra equipped with a symmetric, cyclic, bilinear form , : A ⊗ A → R of degree m ∈ Z satisfying Leibniz rule (see Example 5.9).Then , induces a dg A-bimodule map θ : A → A ∨ [m], and hence a cochain map Θ : CH(A, A) → CH(A, A ∨ [m]) between Hochschild cochains.Let CH cyc (A, A ∨ [m]) be the subcomplex of cyclic invariants in CH(A, A ∨ [m]).Let M be the chain model of the gravity operad that Ward constructed (see also Example 5.3(3)).
Theorem 1.2 (See Corollary 6.8).Given A, , , θ, Θ as above, there is an action of M on Θ −1 (CH cyc (A, A ∨ [m])), giving rise to a structure of a gravity algebra up to homotopy.If θ is a quasi-isomorphism and Θ restricts to a quasi-isomrphism ), this descends to a gravity algebra structure on the cyclic cohomology of A, which is compatible with the BV algebra structure on Hochschild cohomology.
Here the BV algebra structure on the Hochschild cohomology of A (when θ is a quasiisomorphism) is well-known (e.g.Menichi [28,Theorem 18]), where the BV operator is given by Connes' operator (Example 2.6).By compatibility with a BV algebra structure we mean the content of Lemma 5.1.Note that Ward's original theorem only applies to the situation that θ is an isomorphism ([31, Corollary 6.2]).
Chain level structures in S 1 -equivariant string topology.Let us say more about Irie's work [20].Using his chain model and results of Ward ([31, Theorem A, Theorem B]), Irie obtained an operadic chain level refinement of the string topology BV algebra, and compared it with a solution to the ordinary Deligne's conjecture via a chain map defined by iterated integrals of differential forms.
The third result in this paper is a similar story in the S 1 -equivariant context.Note that the string topology BV algebra induces gravity algebra structures on two versions (positive i.e. ordinary, and negative) of S 1 -equivariant homology of LM (Example 7.1).
Theorem 1.3 (See Theorem 7.6)).For any closed oriented C ∞ -manifold M , there exists a chain complex Õcyc M satisfying the following properties.Firstly, the homology of Õcyc M is isomorphic to the negative S 1 -equivariant homology of LM , and Õcyc M admits an action of M (hence an up-to-homotopy gravity algebra structure) which lifts the gravity algebra structure mentioned above.Secondly, there is a morphism of M -algebras which is induced by iterated integrals of differential forms, where the structure on righthand side follows from Theorem 1.2 and Θ comes from the Poincaré pairing.At homology level, the morphism (1.1) descends to a map (part of arrow 4 below) which fits into a commutative diagram of gravity algebra homomorphisms Here A is the S 1 -equivariant homology of LM , B is the negative cyclic cohomology of Ω(M ), C is the negative S 1 -equivariant homology of LM , D is the cyclic cohomology of Ω(M ).Arrows 1, 4 are defined by iterated integrals on free loop space, and arrow 2 (resp.3) is the connecting map in the tautological long exact sequence for S 1 -equivariant homology theories (resp.cyclic homology theories).
The crucial part of Theorem 1.3 is, of course, the chain level statement that fits well with structures on homology.The first part of Theorem 1.3 was conjectured by Ward in [31, Example 6.12], but the correct statement turns out to be more complicated, as we actually lift gravity algebra structures on negative S 1 -equivariant homology rather than S 1 -equivariant homology, whereas they are naturally related by a morphism (arrow 2).
Other than the chain level statement, part of the results at homology level is known.For example, the fact that arrow 1 is a Lie algebra homomorphism appeared in work of Abbaspour-Tradler-Zeinalian as [1,Theorem 11]; The fact that (1.2) commutes was of importance to Cieliebak-Volkov [8] (the arrows are only treated as linear maps there).
In a forthcoming paper, the author is going to apply results in this paper to Lagrangian Floer theory, in view of cyclic symmetry therein (Fukaya [11]).
Outline.In Section 2, we review cyclic homology of mixed complexes.In Section 3, we prove Theorem 1.1.In Section 4, we review Irie's de Rham chain complex of differentiable spaces and apply Theorem 1.1 to it.In Section 5, we review basics of operads and algebraic structures.In Section 6, we prove Theorem 1.2.In Section 7, we prove Theorem 1.3.

Conventions.
Vector spaces are over R, algebras are associative and unital, graded objects are Z-graded.Homological and cohomological gradings are mixed by the understanding As for sign rules, see Appendix A. For the sake of convenience, we may write (−1) ε for a sign that is apparent from Koszul sign rule (Appendix A.1).
These are three classical versions of cyclic homology of mixed complexes, called the negative, periodic and ordinary (positive) cyclic homology of (C * , b, B), respectively.We prefer to distinguish them by suggestive symbols ( ) rather than names, as did in [8].Here cohomological grading is used for cyclic homology since we deal with cochain complexes.If we move to homological grading C * := C − * and replace u by v (a formal variable of degree −2), then the mixed chain complex (C * , b, B) gives negative, periodic and ordinary (positive) cyclic homology theories 8] also takes the Hom dual of C to define cyclic cohomology theories of (C, b, B), which we try to avoid in this article.) For any mixed cochain complex (C * , b, B), there is a tautological exact sequence The connecting map B 0 * : HC Similarly, from the short exact sequences one obtains the Gysin-Connes exact sequences The connecting maps ) and the exact sequences (2.2) fit into the following commutative diagram: Proof.The left and the right squares commute since they commute at the level of cocycles.
As for the middle square, let (1) A series of linear maps {f i : A morphism between mixed complexes is an ∞-morphism {f i } i≥0 such that f i = 0 for all i > 0, namely a single degree 0 linear map that commutes with both b and B. A quasi-isomorphism between mixed complexes is a morphism that is also a (b, b ′′ )-quasiisomorphism.A homotopy between two morphisms f, g : (C * , b, B) → (C ′′ * , b ′′ , B ′′ ) is an ∞-homotopy {h i } i≥0 such that h i = 0 for all i > 0, namely a single degree −1 linear map h satisfying f − g = b ′′ h + hb and B ′′ h + hB = 0.
The following important lemma goes back to [21, Lemma 2.1], and is a special case of [34,Lemma 2.3] which is stated for S 1 -complexes (an ∞-version of mixed complexes).The proof is a spectral sequence argument using the u-adic filtration on C . The following lemma illustrates the naturality of the tautological exact sequence and Connes-Gysin exact sequences for cyclic homology, with respect to ∞-morphisms between mixed complexes.
) be an ∞-morphism.Then f = i u i f i induces a morphism between the exact sequence (2.1) for C and C ′′ , namely there is a commutative diagram Similarly, for the exact sequence (2.2a), there is a commutative diagram The case of the exact sequence (2.2b) is also similar.
Proof.We only write proof for the first diagram since the others are similar.The left and the middle squares commute since they commute at the level of cocycles.Now let c = Using these relations, it is a straightforward computation to see , and the right-hand side is exact, so commutativity of the right square is proved.
We now discuss some important examples of mixed (co)chain complexes and their cyclic homologies.Recall that a cosimplicial object in some category is a sequence of objects C(k) (k ∈ Z ≥0 ) together with morphisms satisfying the following relations: A cocyclic object is a cosimplicial object {C(k)} k together with morphisms τ k : C(k) → C(k) satisfying the following relations: be the standard simplices, then {∆ k } k∈Z ≥0 is a cocyclic set (topological space, etc.) with standard cocyclic maps δ Example 2.6 (Cocyclic complex and Connes' version of cyclic cohomology).Consider the category of cochain complexes where the morphisms are degree 0 cochain maps.Let (C(k) * , d), δ i , σ i , τ k be a cocyclic cochain complex, then a mixed cochain complex (C, b, B) is obtained as follows.Let then δ 2 = 0, δd + dδ = 0. Let (C * , b) be the product total complex of the double complex C(k) l , d, δ k∈Z ≥0 l∈Z : For later purpose we also introduce the normalized subcomplex Note that the natural inclusion where λ, N, s are given by (here |c| is the degree of , so Connes' operator B has simpler form on normalized subcomplex: forms a subcomplex (we denote this inclusion by i λ ).This leads to Connes' version of cyclic cohomology of the cocyclic cochain complex, . By an argument similar to [25, Theorem 2.1.5,2.1.8]one sees that this inclusion The short exact sequence 0 Here we have made use of an isomorphism and the fact that (C, b ′ ) is acyclic (since b ′ s + sb ′ = 1).Lemma 2.9 below says (2.6) can be identified with (2.2b).Finally we mention that HC * , where the first isomorphism follows from Lemma 2.3.
A subexample of Example 2.6 is as follows.
Example 2.7 (Cyclic cohomology of dg algebras).Let A * be a dg algebra with unit 1 A .Then {Hom * (A ⊗k+1 , R)} k≥0 has the structure of a cocyclic cochain complex, where δ i : (2.8) The associated mixed total complex is denoted by CH * (A, A ∨ ).For simplicity, denote cyclic homologies of Let us also recall that for any dg A-bimodule M * , there is a structure of a cosimplicial complex on {Hom * (A ⊗k , M )} k≥0 , where δ i : Hom * (A ⊗k−1 , M ) → Hom * (A ⊗k , M ), The associated total complex, denoted by CH * (A, M ), is called the Hochschild cochain complex, whose cohomology group, denoted by HH * (A, M ), is called the Hochschild cohomology.
one sees that the cosimplicial structure on {Hom * (A ⊗k , A ∨ )} k is the same as that on {Hom * (A ⊗k+1 , R)} described previously, in view of the natural isomorphism from Hom-⊗ adjunction.See Example 5.9 for further discussion.
Remark 2.8.We shall use the name "Connes' version of cyclic cohomology" for "cocyclic complex", even if we work with chain complexes rather than cochain complexes.For a cocyclic chain complex . Lemma 2.9.In the situation of Example 2.6, the isomorphism (2.5) and the long exact sequences (2.6) and (2.2b) fit into the following commutative diagram: Proof.The left square commutes since it commutes at cochain level.To verify commutativity of the other two squares, we need explicit formulas of B λ and S λ .Since where by examining (2.7), Let us calculate that on Z(C, b), , which says the middle square commutes.Similarly, S λ is the composition where R λ * : (2.10) also holds on Z(C/C cyc , b), and implies (1 e. the right square commutes.Example 2.10 (S 1 -equivariant homology theories [21]).Let X be a topological S 1 -space, namely a topological space with a continuous S 1 -action F X : S 1 ×X → X.Let (C * , b) = (S * (X), ∂) be the singular chain complex of X, and define the rotation operator B = J : Here and × is the simplicial cross product induced by standard decomposition of To see J 2 = 0, let us write down the cross product with [S 1 ] explicitly.For k ∈ Z ≥0 and j ∈ {0, . . ., k}, consider the embeddings ι k,j : ∆ k+1 → ∆ 1 × ∆ k defined by ι k,j (t 1 , . . ., t k+1 ) := (t j+1 , (t 1 , . . ., t j , t j+2 , . . ., t k+1 )), we conclude that for any a ∈ S * (X), ) is isomorphic to the S 1 -equivariant homology of X, i.e. homology of the homotopy quotient (Borel construction).The other two cyclic homology groups of (S * (X), ∂, J) are called the negative and periodic S 1 -equivariant homology of X, and are denoted by We end this example by mentioning that (2.13a) coincides with the Gysin sequence associated to the Remark 2.11.There seems to be no interpretation of G S 1 * (X) and H S 1 * (X) as homology groups of some spaces naturally associated to X, but there are homotopy theoretic interpretations.For example, when X is a (finite) * (X) is naturally isomorphic to the homotopy groups of the homotopy fixed point spectrum (H ∧ X + ) hS 1 , where H is the Eilenberg-MacLane spectrum {K(Z, n)}.

A COCYCLIC COMPLEX AND AN ∞-QUASI-ISOMORPHISM
Let X be a topological space with S 1 -action Taking singular chains of the cocyclic space {X ×∆ k } k≥0 yields a cocyclic chain complex {S * (X × ∆ k )} k≥0 .Let us denote the associated mixed complex by The S 1 -action on X extends to X × ∆ k where the S 1 -action on ∆ k is trivial, and then the rotation operator J : S * (X) → S * +1 (X) defined in Example 2.10 extends component-bycomponent to S X∆ * by . By Example 2.10, J 2 = 0 and ∂J + J∂ = 0. Since S 1 acts trivially on ∆ k , J commutes with δ i , σ i .It follows that δJ + Jδ = 0 and J(S X∆,nm ) ⊂ S X∆,nm , so (S X∆ * , b, J), (S X∆,nm * , b, J) are also mixed complexes.J also commutes with τ X×∆ k k because of the commutative diagram so JB + BJ = 0. We will analyze the relationship between the mixed complexes If there is no risk of confusion, we shall write and the induced maps on singular chain complexes as δ i , σ i , τ k for short.Note that δ i , σ i do not involve S 1 -action, so if we forget the S 1 -action, there is still a total complex Let us state the main theorem of this section.
Theorem 3.1.Let X be a topological S 1 -space.Then for both of the mixed complex structures (b, B) and (b, J) on S X∆ * = k≥0 S * +k (X × ∆ k ), there are natural isomorphisms Proof.The statement about isomorphisms is a consequence of Lemma 2.3, Corollary 3.5 and Proposition 3.7 below.The statement about long exact sequences is then a consequence of Lemma 2.4.Corollary 3.2.For any topological S 1 -space X, Connes' version of cyclic cohomology of the cocyclic chain complex {S * (X × ∆ k )} k∈Z ≥0 is naturally isomorphic to the negative S 1 -equivariant homology of X. Lemma 3.3.For any topological space X, the projection chain map Proof.Since pr 0 is surjective, it suffices to prove Ker(pr 0 Let us write S * := Ker(pr 0 ) * and consider the decreasing filtration F p (p ∈ Z ≥1 ) on S defined by F p S * := k≥p S * +k (X × ∆ k ).The E 1 -page of the spectral sequence of this filtration is divided into columns indexed by q ∈ Z ≥0 , each of which looks like For each k ≥ 1, the map x is a homotopy equivalence.Since p k+1 = p k σ k and σ j δ i = id (i = j, j + 1), we conclude that for any k ≥ 1 and Thus all E 2 -terms vanish.Remark 3.4.The proof of Lemma 3.3 implies that more generally, for a cosimplicial complex Corollary 3.5.For any topological S 1 -space X, pr 0 : (S X∆ * , b, J) → (S * (X), ∂, J) is a mixed complex quasi-isomorphism.
Lemma 3.6.Consider the topological S 1 -space S 1 with rotation action on itself.
(1) Choose ξ = {ξ n } n≥0 as in Lemma 3.6 (1).Define a sequence of linear maps , so we omit it.Let us write where the last equality follows from b(ξ n ) = (J − B)(ξ n−1 ).Now introduce maps It follows that . Since pr 0 is a quasiisomorphism, so is f ξ 0 .

THE STORY OF DIFFERENTIABLE SPACES
4.1.Differentiable spaces and de Rham chains.Materials in this subsection are collected from Irie [20].The notion of differentiable spaces is a modification of that utilized by Chen [6], and the notion of de Rham chains is inspired by an idea of Fukaya [10].
Let U := n≥m≥0 U n,m where U n,m denotes the set of oriented m-dimensional C ∞submanifolds of R n .Let X be a set.A differentialble structure P(X) on X is a family of maps {(U, ϕ)} called plots, such that: • Every plot is a map ϕ from some U ∈ U to X; A differentiable space is a pair of a set and a differentiable structure on it.A map f : X → Y between differentiable spaces is called smooth, if (U, f • ϕ) ∈ P(Y ) for any (U, ϕ) ∈ P(X).A subset of a differentiable space and the product of a family of differentiable spaces admit naturally induced differentiable structures ([20, Example 4.2(iii)(iv)]).
Remark 4.1.Differentiable structures are defined on sets rather than topological spaces.
For later purpose, we say a differentiable structure and a topology on a set X are compatible if every plot is continuous.
Example 4.2.Here are some important examples of differentiable spaces.
(1) Let M be a C ∞ -manifold.Consider two differentiable structures on it: (a) Define (U, ϕ) The set-theoretic identity map id M : M reg → M is smooth, but its inverse is not.(2) Let LM := C ∞ (S 1 , M ) be the smooth free loop space of M , where There is a differentiable structure P(L M ) on LM defined by: (U, ϕ) ) For each k ∈ Z ≥0 , the smooth free loop space of M with k inner marked points, denoted by L k+1 M , is defined as ) as a subspace of L M ×∆ k , where ∆ k is viewed as a subspace of R k with the differentiable structure in (1a).(4) The smooth free Moore path space of M , denoted by ΠM , is defined as • The map ) and the map U → M , u → ϕ γ (u)(t 0 ) is a submersion for t 0 = 0, T .(5) For each k ∈ Z ≥0 , the smooth free Moore loop space of M with k inner marked points, denoted by L k+1 M , is defined as Apparently there are two ways to endow the set L k+1 M with differentiable structures, namely as a subset of (ΠM ) k+1 or of ΠM × R k .It basically follows from [20, Lemma 7.2] that these two ways are equivalent.Let us denote by L M k+1 (resp.L M k+1,reg ) the differentiable space obtained from Π M (resp.Π M reg ).Note that the inclusion of sets L k+1 M = {T = 1} ⊂ L k+1 M , induced by the inclusion The de Rham chain complex (C dR * (X), ∂) of a differentiable space X is defined as follows.For n ∈ Z, let CdR n (X) := (U,ϕ)∈P(X) Ω dim U−n c (U ).For any (U, ϕ) ∈ P(X) and ω ∈ Ω dim U−n c (U ), denote the image of ω under the natural inclusion n (X) be the subspace spanned by all elements of the form and π : U ′ → U is a submersion.Then define C dR n (X) := CdR n (X)/Z n .By abuse of notation we still denote the image of (U, ϕ, ω) under the quotient map CdR n (X) → C dR n (X) by (U, ϕ, ω).Then ∂ : Remark 4.3.For any oriented C ∞ -manifold M , there exists n ∈ Z ≥0 and an embedding ι : M ֒→ R n .Then (ι(M ), ι −1 ) ∈ P(M reg ) ⊂ P(M ), and . Such a de Rham chain is independent of choices of n and ι, and by abuse of notation we write it as (M, id M , ω).If M is closed oriented, we call (M, id M , 1) the fundamental de Rham cycle of M (or M reg ).
Let X, Y be differentiable spaces.The cross product on de Rham chains is a chain map 4.2.S 1 -equivariant homology of differentiable S 1 -spaces.Let X be a differentiable S 1 -space, namely X is a differentiable space with a smooth map where S 1 is endowed with the differentiable structure in Example 4.2(1a).Let (S 1 , id S 1 , 1) ∈ C dR 1 (S 1 ) be the fundamental de Rham 1-cycle of S 1 .Define J : C dR * (X) → C dR * +1 (X); a → F X * ((S 1 , id S 1 , 1) × a), then J is clearly an anti-chain map.We claim J 2 = 0. Let g : S 1 × S 1 → S 1 be the smooth map then by the same arguments as in Example 2.10, to see ).This is easy, as we can see the following: The middle equality on the second line holds since g is a submersion.Thus (C dR * (X), ∂, J) is a mixed chain complex.One can then define the positive (ordinary), periodic and negative "S 1 -equivariant de Rham homology" of X as the versions of cyclic homology of (C dR * (X), ∂, J).Consider ∆ k as a differentiable subspace of R k .Then the cocyclic maps δ i , σ i , τ k among {X × ∆ k } k∈Z ≥0 , defined by the same formulas as in Section 3, are smooth maps between differentiable spaces.So {X × ∆ k } k∈Z ≥0 is a cocyclic differentiable space and {C dR * (X × ∆ k )} k∈Z ≥0 is a cocyclic chain complex, which gives rise to a mixed complex The smooth S 1 -action S 1 × X → X also extends trivially to S 1 × X × ∆ k → X × ∆ k and gives a mixed complex (C X∆ * , b, J).There is counterpart of Theorem 3.1 for differentiable S 1 -spaces, whose proof is also similar.We omit the details since we will not make essential use of it.
The smooth singular chain complex (C sm * (X), ∂) of a differentiable space X, introduced in [20,Section 4.7], is defined in a similar way as the singular chain complex of topological spaces, except that only "strongly smooth" maps ∆ k → X are considered.The homology of (C sm * (X), ∂) is denoted by H sm * (X).Smooth singular homology is related to singular homology and de Rham homology in the following way.
• Let X be a differentiable space with a fixed compatible topology (Remark 4.1).
Then every strongly smooth map , there is a natural transformation The homotopy class of ι u (X) does not depend on u (since H dR,pt By Lemma 3.
Proposition 4.8.For any closed oriented C ∞ -manifold M , there are natural isomorphisms with cocyclic maps, inducing a zig-zag of mixed complex morphisms between the mixed total complexes associated to the cocyclic de Rham chain complexes of these cocyclic differentiable spaces.By Lemma 4.7, this is a zig-zag of mixed complex quasi-isomorphisms.The rest is obvious in view of Lemma 2.3, Proposition 4.5 and Example 4.6.

PRELIMINARIES ON OPERADS AND ALGEBRAIC STRUCTURES
Let V = {V i } i∈Z be a (homologically) graded vector space.A Lie bracket of degree n ∈ Z is a Lie bracket on V [−n], namely a bilinear map [, ] : V ⊗ V → V of degree n satisfying shifted skew-symmetry and Jacobi identity: Note that in this definition, there is no need to apply sign change (A.1).
A structure of Gerstenhaber algebra is a Lie bracket of degree 1 and a graded commutative (and associative, by default) product • satisfying the Poisson relation: A structure of Batalin-Vilkovisky (BV) algebra is a graded commutative product • and a linear map ∆ : V * → V * +1 (called the BV operator) such that ∆ 2 = 0, and By induction, the defining relation (5.1) implies that for any k ≥ 2, where ε(i, j) is from the Koszul sign rule.By [15, Proposition 1.2], a BV algebra is equivalently a Gerstenhaber algebra with a linear map ∆ : Following Getzler [16], a structure of gravity algebra is a sequence of graded symmetric linear maps . ., a k } (which we call k-th bracket), satisfying the following generalized Jacobi relations: Note that the relation for (k, l) = (3, 0) implies that, with sign change (A.1), the second bracket becomes an honest Lie bracket on The following lemma, which goes back to [3, Theorem 6.1], is well-known to experts.
Lemma 5.1.Let (V * , •, ∆) be a BV algebra, W * be a graded vector space, with linear maps α : (1) W * is a gravity algebra where the brackets W ⊗k → W are (2) Let [, ] be the Gerstenhaber bracket (5.3) on V * .Then for any Proof.To prove (1), first note that since • is graded commutative, {x 1 , . . ., x k } is graded symmetric in its variables.Next, the generalized Jacobi relations follow from a straightforward calculation based on (5.2) (see the proof of [7, Theorem 8.5]), and is omitted.The proof of ( 2) is trivial.
A BV algebra homomorphism between two BV algebras is an algebra homomorphism that commutes with their BV operators.The case of gravity algebras is similar.The following lemma is obvious.Lemma 5.2.Suppose there is a commutative diagram of linear maps ) satisfy the assumptions in Lemma 5.1, and g is a BV algebra homomorphism.Then f is a gravity algebra homomorphism (for the induced structures on W ,W ′ ).
Next we need to work in the language of operads.We collect some basics below, and refer the reader to [20, Section 2] or standard references [26][27] for more details.
Let (C , ⊗, 1 C ) be a symmetric monoidal category.A nonsymmetric operad (ns operad for short) O in C consists of the following data: (5.5b) , which a two-sided unit for • i .
An operad is a ns operad such that each O(k) admits a right action of the symmetric group S k (S 0 is the trivial group), in a way compatible with partial compositions.A (ns) operad in the symmetric monoidal category of dg (resp.graded) vector spaces is called a (ns) dg (resp.graded) operad.A Koszul sign (−1) |y||z| should appear in (5.5b) in graded and dg cases.Taking homology yields a functor from the category of (ns) dg operads to the category of (ns) graded operads.
Example 5.3.Here are some examples of dg operads and graded operads.
(1) (Endomorphism operad End V .)For any dg (resp.graded) vector space V * , there is a dg (resp.graded) operad End V defined as follows.For each k ≥ 0, Let O be a (ns) graded operad or dg operad.A structure of algebra over O on V , or say an action of O on V , means a morphism O → End V as (ns) operads.(2) (Gerstenhaber operad Ger, BV operad BV, and gravity operad Grav.)These are graded operads that can be defined in terms of generators subject to the relations defining Gerstenhaber/ BV/ gravity algebras.A Gerstenhaber/ BV/ gravity algebra is exactly an algebra over Ger/ BV/ Grav.(3) (Ward's construction [31].)There is a dg operad M constructed from certain "labeled A ∞ trees", such that H * (M ) ∼ = Grav as graded operads, and there are explicit homotopies measuring the failure of gravity relations on M (while Jacobi relation for the second bracket strictly holds).For this reason an algebra over M can be viewed as a gravity algebra up to homotopy.M is closely related to the operad of "cyclic brace operations" (Section 6).There are other important properties of M that we will use later (Proposition 5.6( 5)).Indeed the notation of Ward [31] is M , but we use M for the reason of Remark 5.5.
Definition 5.4.([20, Definition 2.6, 2.9].)Let O be a ns dg operad. ( , and that for any Remark 5.5.An operad with a cyclic structure is called a cyclic operad.The cyclic relation in Definition 5.4 differs from some authors (in particular, Ward [31]) in the orientation of performing cyclic permutation, but they are equivalent.See e.g.[28,Section 3].
Let O = (O(k)) k≥0 be a ns dg operad endowed with a multiplication µ and a unit ε.Remark 5.11.Statement (3) ′ in Example 5.9 is irrelevant to the algebra structure on A. It holds true when A is just a graded vector space endowed with a symmetric bilinear form , : A×A → R of degree m.In this case, we shall write )).

CYCLIC BRACE OPERATIONS
This section is devoted to the proof of Theorem 1.
where the summation is taken over tuples (i 1 , . . ., i n ) ∈ Z n ≥1 satisfying i j+1 ≥ i j + t j and i n ≤ r − n + 1 + n−1 l=1 t l .The sign ± is from iteration of (A.2).Brace operations were first described by Getzler [14] in Hochschild context (generalizing the Gerstenhaber bracket [12] which corresponds to n = 2) and later by Gerstenhaber-Voronov [13] in operadic context.There is also an interpretation of brace operations via planar rooted trees, going back to the "minimal operad" of Kontsevich-Soibelman [24] (see also [9,), which allows for a variation in the cyclic invariant setting ( [31]).
Let us fix terminologies about trees before moving to more details.
• A tree without tails is a contractible 1-dimensional finite CW complex.A 0-cell is called a vertex; the closure of a 1-cell is called an edge (identified with [0, 1]).• A tree with tails is a tree without tails attached with copies of [0, 1) called tails by gluing each 0 ∈ [0, 1) to some vertex.The set of vertices, edges and tails in a tree T is denoted by V T , E T and L T , respectively.The set of edges and tails at v ∈ V T is denoted by E v and L v , respectively.The valence of a vertex v is the number The arity of a vertex is its valence −1.
• An oriented tree is a tree with a choice of direction for each edge, from one vertex to the other.Such a choice of directions is called an orientation of the tree.• A rooted tree is a tree with a choice of a distinguished tail called the root.
Every rooted tree is naturally oriented by directions towards the root.
• A planar tree is a tree with a cyclic order on E v ∪ L v for each vertex v.
Every planar tree can be embedded into the plane in a way unique up to isotopy, so that at each vertex v, the cyclic order on E v ∪ L v is counterclockwise.
Every planar rooted tree T carries a natural total order on E T ∪ L T , which can be obtained by moving counterclockwise along the boundary of a small tubular neighborhood of T in the plane.It starts from the root and is compatible with the cyclic order on E v ∪ L v for each v ∈ V T , and also restricts to total orders on E T , L T and E v , L v for each v ∈ V T .
• An n-labeled tree is a tree T with a bijection between {1, 2, . . ., n} and V T .If the number of vertices is not specified, it is just called a labeled tree.The vertex with label i in an n-labeled tree T is denoted by v i (T ), with arity a i (T ).The notion of isomorphisms of trees (with various structures) is obvious.We shall view isomorphic trees as the same.
For n ∈ Z ≥1 , let B s (n) be the set of n-labeled planar rooted trees without non-root tails, and let B(n) be the vector space spanned by B s (n).Let Bs (n) be the set of n-labeled planar rooted trees with tails, and let B(n) be the vector space spanned by Bs (n).
Given as follows.κ n is the restriction of κn .For T ′ ∈ Bs (n) and where by convention summation over the empty set is zero.
In [31, Section 3.2], Ward introduced an operad B which he called cyclic brace operad.Let B s (n) be the set of oriented n-labeled planar trees without tails.Then B (n) is the graded vector space spanned by B s (n) modulo the relation that reversing direction on an edge produces a negative sign.If there is no risk of confusion, we will by abuse of notation not distinguish T ∈ B s (n) from its image in B (n).There is a morphism of operads ρ : B → B, which is induced by maps where R 1 (T ) is the set of labeled planar rooted trees that can be obtained by adding a root to the (non-rooted) tree underlying T , and ε(T , T ) is the number of edges in E T = E T whose direction from T does not agree with the direction from the rooted structure of T .
Here and hereafter, in appropriate contexts we use f s to denote a set-theoretic map which induces a linear map f .A natural example of cyclic brace algebras, i.e. algebras over B , is as follows.Restricting (κ • ρ)(T ) to Õcyc gives an element in Hom(( Õcyc ) ⊗n , Õ).Moreover, if (Such a claim appears in [31,Theorem 5.5] where it is referred to [31, Proposition 3.10], but there is no direct proof given in [31].We will give a direct proof in a slightly different situation.)Hence κ • ρ gives a morphism B → End Õcyc .Definition 6.4 (Cyclic brace operations).Let O be a dg cyclic operad.The cyclic brace operations on Õcyc are those characterized by the linear maps discussed in Example 6.3.
Remark 6.5.It seems hard to write a direct formula for cyclic brace operations on Õcyc in terms of operadic compositions, in a way as explicit as (6.1).
Consider Bs (n) ⊃ B s (n) and B (n) ⊃ B (n) by extending the definitions to labeled planar trees with tails.There is a forgetful map w s n : Bs (n) → Bs (n) \ B s (n) forgetting the choice of root but keeping the orientation from rooted structure.Note that w s n induces w n : B(n) → B (n)/B (n).There is also a map r s n : Bs where R 0 (T ′ ) is the set of n-labeled planar rooted trees obtained by choosing one of the tails in T ′ as the root, and ε(T ′ , T ′ ) is defined similar to ε(T , T ) in (6.4).It is clear that To describe r n • w n , consider a map t s n : Bs (n) → Bs (n) so that T ′ and t s n (T ′ ) are the same after forgetting the root, and the root of t s n (T ′ ) is the first non-root tail of T ′ (if there are no non-tail roots then t s n (T ′ ) = T ′ ).Then for any Here ε(T ′ , t i n (T ′ )) is the number of edges in E T ′ = E t i n (T ′ ) whose direction towards the root of T ′ does not agree with the direction towards the root of t i n (T ′ ).
where T (k, T ) ⊂ Bs (n) is defined similar to T (k, T ) in (6.2).
Lemma 6.6.Let O be a dg operad.For any T ∈ B s (n) and Proof.Consider the set of labeled planar rooted trees whose vertices have arities equal to k = (k 1 , . . ., k n ) in accordance with the labeling.Such a set can be represented as and the result follows.
In the rest of this section, we take O = End A , where A is a dg algebra endowed with a symmetric, cyclic, bilinear form , of degree m.Recall from Example 5.9 that , induces θ : A → A ∨ [m] and Θ : CH(A, A) → CH(A, A ∨ [m]).To deal with signs, we may work with A[1] instead of A. As explained in Appendix A, the symmetric bilinear form , on A becomes anti-symmetric on A [1], and the cyclic permutation τ k on Hom(A ⊗k+1 , R) reads as τk = (−1) , there is no need to take operadic suspension of B , and B (n) stands in degree 0 when dealing with A [1].
Since the pairing , is not necessarily nondegenerate, there is not always a cyclic structure on End A compatible with cyclic permutations on {Hom(A ⊗k+1 , R)} via the map Hom(A ⊗k , A) → Hom(A ⊗k , A ∨ [m]) induced by , , so the discussion of Example 6.3 does not directly apply here.However, the following is true.Proposition 6.7.There is a natural action of B on Θ −1 ( k≥0 Hom * +m cyc (A ⊗k+1 , R)).
Proof.(This proposition is irrelevant to the multiplication on A; compare Remark 5.11.)Similar to Example 6.3, it suffices to show if T ∈ B s (n) and f i ∈ Hom(A ⊗ki , A) is weakly cyclic invariant in the sense that λ(θ is weakly cyclic invariant.This is immediate from Lemma 6.6 and Lemma 6.9 below. A similar application of Lemma 5.1 to a part of the Connes-Gysin sequence (2.2b) for the mixed complex (S * (LM ), ∂, J), together with Lemma 2.1 and Lemma 5.2, yields the following lemma.Lemma 7.2.For any closed oriented manifold M , there is a gravity algebra structure on G S 1 * +dim M (LM ), such that the natural map 12) is a morphism of gravity algebras.where evj × ev0 is the fiber product of de Rham chains with respect to evaluation maps ev j : L M k+1,reg → M reg and ev 0 : L M k ′ +1,reg → M reg (it is well-defined because of submersive condition), and con j : L k+1 M evj × ev0 L k ′ +1 M → L k+k ′ M is the concatenation map defined by inserting the second loop into the first loop at the j-th marked point.
Note that S * (X) = S X∆ * (0) = S X∆,nm * (0) by vacuum normalized condition.Since (S X∆,nm * , b) ֒→ (S X∆ * , b) is a quasi-isomorphism, Lemma 3.3 and Corollary 3.5 also hold true if S X∆ * is replaced by S X∆,nm * .In the following, we may use S X∆,nm * to simplify calculation involving Connes' operator B. One could also stick with S X∆ * , though.Recall the augmentation map ε 3 and Remark 3.4, all vertical arrows are quasi-isomorphisms, and by assumption, the arrows in the second row are quasi-isomorphisms.Thus the arrows in the first row are quasi-isomorphisms.In this way we obtain quasi-isomorphisms of mixed complexes (S X∆ * , b, B) ← (C sm,X∆ * , b, B) → (C dR,X∆ * , b, B), and get the desired isomorphisms by Lemma 2.3, Theorem 3.1 and Corollary 3.2.Compatibility with long exact sequences is a consequence of Lemma 2.4 and Lemma 2.9.Example 4.6.Let M be a closed oriented C ∞ -manifold.It is proved in [20, Section 5, Section 6] that Assumption 4.4 is satisfied for M, M reg (with manifold topology) and L M (with Fréchet topology) in Example 4.2.Moreover, L M is an S 1 -space that Propositon 4.5 applies to.